On accounting for tempering effect in unsteady creep model for metals

. A mathematical model describing processes of nonstationary creep of metals under multiaxial stress state is developed. A phenomenon of reverse creep modeling is considered in more details. The results of numerical modeling of the reverse creep process in variety of structural steels are presented. Obtained numerical results are compared with available results of physical experiments.


Introduction
The study of the laws of inelastic deformation of structural materials under uniaxial and multiaxial stressed state is important for the development of the fundamentals of the mathematical theory of creep. With a considerable number of experimental studies of the creep process of structural alloys [1], experiments on the study of the phenomenon of reverse creep (the phenomenon of decreasing deformations in time after removing the load) of metals is comparatively small. As a result, this effect is not taken into account in most cases during the calculation of structural elements creep nowadays.
At the same time, in order to verify the physical reliability of the defining equations of the creep theory, calculations and comparison of the obtained numerical results with the data available in the literature on a wide range of experiments are required [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The results of such researches can also be useful in determining (specifying) the material parameters of the scalar functions and functionals involved in defining equations of nonstationary creep. Moreover, fast unloading of elements and components of load-bearing structures can lead to stress loading of the opposite sign, which in some cases is unacceptable (deformation of reverse creep is commensurable with elastic deformation).
A mathematical model of the nonstationary creep of metals under a multiaxial stressed state was developed in [17][18][19]. In [19], reliability estimation was carried out and the boundaries of applicability of the developed defining relations of nonstationary creep under multiaxial stress state were determined. Applicability boundaries of the developed defining relations of nonstationary creep under multiaxial stress state were determined as well as reliability estimation was done in [19].
This model is modified to describe the phenomenon of inverse creep and an estimate of its reliability by comparing the obtained numerical results with the experimental data available in the literature on the inverse creep of structural steels is given in this paper.

The defining relations of nonstationary creep
General considerations of the nonstationary creep equations [17][18][19] are: -Initially isotropic media are considered.
-The strain and strain rates tensors are sums of the "instantaneous" and " time-dependent" components. The "instantaneous" component consists of elastic strains that do not depend on the loading history and determined by the final state of the process, and plastic strains depending on the loading history. The time component (creep strains) describes the time dependence of deformation processes at low loading speeds.
-The equipotential surface evolution is determined by changes in its radius с С and displacements of its The relationship between the stress tensor and elastic strain tensor is based on thermoelasticity equations: where  , e are spherical and ij   , ij e are deviatoric components of the corresponding stress ij  and strain ij e tensors; ( ) G T is the shear modulus, ( ) is the coefficient of linear temperature expansion of the material.
To describe the creep processes, we introduce in the stress space the equipotential creep surfaces c F having the same centre c ij  and different radii c C , determined by the current state of stress: In accordance with the associativity law where c  corresponds to the current surface The surface with radius c С may be outlined among these equipotential surfaces as corresponding to zero creep rate: where с С and c  are experimentally determined functions of temperature Т. The evolution equation for the coordinates of the centre of the creep surface has the following form [17][18][19]: where 1 c g and 2 c g > 0 are experimentally determined material parameters.
By specifying (3) the law of gradientality can be represented as The intensity of the creep strain rate tensor has the following form: By taking into account (8) we are getting the following expression for the length of the creep strain path: and   The dependence of с  on the time t with c u S const  in the case of multi-axis straining over the ray trajectory is shown in Fig. 1. The curve с t  (Fig. 1) can be conventionally divided into three segments: On the segment ( (1) 11 11 11 , when 0, , when .
Taking into account these equations we may represent I с  on the first segment of creep curve as: where     (0) 11 are the values of с  in points "0" and "1", respectively.
On third segment ( (2) The analysis for straining process of the specimen using the model of nonstationary creep was performed with 35 steel material parameters shown in Table 1.
The test results and their comparison with the obtained numerical data are shown in Fig. 2, which shows the creep curve (the solid line indicates the numerical results obtained by authors, and the markers indicate the corresponding experimental data). Qualitative and quantitative coincidence of experimental and calculated data observed. the same with previous example). The analysis of the straining process of specimen using the model of nonstationary creep was performed with ATV steel material parameters shown in Table 1.
The test results and their comparison with the obtained numerical data are shown in Fig. 3, which shows the creep curve (the solid line indicates the numerical results obtained by authors, and the markers indicate the corresponding experimental data). Qualitative and quantitative coincidence of experimental and calculated data observed here as well.

Conclusions
A mathematical model describing the processes of nonstationary creep of structural materials (metals and their alloys) under complex loading is developed. The results of numerical simulation and experimental data of reverse creep processes of structural steels are compared. It is shown that the developed variant of the defining relations of nonstationary creep allows to describe the processes of reverse creep of metals with required for engineering calculations accuracy.